Numerical Solution Of Adaptive Wavelet Neural Networks For The Second Kind Of Fredholm Integral Equation

As an important subject of mathematics,integral equations' research has been the focus of many scholars all the time.However,it is a pity that,most of the existing numerical methods for solving integral equations have disadvantages such as large amount of complicated calculations,unstable accuracy and slow operation speed.Neural network can achieve good mapping for nonlinear problems,and it can also conduct high-precision parallel processing for multi-dimensional input vectors by self-learning.Because there are translation factor and scaling factor containing within wavelet analysis,so it can analyze the characteristics of data at different scales.Therefore,in order to take advantages of both of them effectively,a wavelet neural network is constructed by combining the wavelet analysis and neural networks.In this paper,the wavelet neural network is used to solve the numerical solution of the second type of Fredhlom integral equation.The first chapter introduces the development background as well as research significance of integral equation and neural network.And outline the neural network field in solving the numerical solution of the integral equation and the wavelet neural network's research status at home and abroad.The second chapter summarizes the basic knowledge and theory needed in this paper,namely neural network and required wavelet neural network.In the third chapter,according to the characteristics of the data itself,"adaptive" suitable wavelet base was selected.The excitation functions in the hidden layer are transformed into a wavelet generating functions which will be suitable for neural network,and the weight and threshold from the input layer to the hidden layer were constructed by using the selected wavelet basis.Therefore,an adaptive wavelet neural network is proposed to solve the numerical solution of one-dimensional second-class Fredhlom integral equation.Finally,the relevant parameters are adjusted adaptively according to the gradient descent algorithm.The fourth chapter regards the three-layer feedforward wavelet neural network as the main research object,and the integral term is expanded by the Nystrom solution of the mean value theorem,combining with self-adaption wavelet neural network to solve the two-dimensional Fredhlom integral equation of the second kind.The fifth chapter summarizes the above overall work,and some suggestions for future research are also given.